Cohomology of product spaces from a categorical viewpoint

Abstract

One of the aims of Algebraic Topology is to study complex problems within Topology by translating them into the more workable world of Algebra. This is usually done by defining invariants such as singular homology and singular cohomology. Both of them help us find properties of spaces. In many cases, homology is not enough and it is desirable to know the cohomology of spaces. For instance, although a product of two Klein Bottles and ($S^1 \vee \mathbb {R}P^2) \times (S^1 \vee \mathbb {R}P^2$) have isomorphic homology modules, their respective cohomologies have different ring structures, as shown in Example 3.15, which can also be found in [8]. All other examples and counterexamples given in this project are due to the author.

These invariants are very powerful since they contain much information of the spaces that are being studied. The first goal of the present work is to prove a formula that relates the homology of a product of spaces with the homologies of the factors. This formula was obtained in the first half of the past century after the work of H. Künneth, who found in 1923 a relation between the Betti numbers of the product of two spaces and the Betti numbers of each of its factors; see [3, Chapter II, Section 5]. In this work, we want to study Künneth formulas for both homology and cohomology. Although the homology Künneth formula is a very standard result, the cohomology Künneth formula is harder to be found in modern literature, and this is why its study is interesting.